If the number has \(‘n’\) distinct prime factors, then number of ways in which the number can be represented as product of two co–primes \( = {\rm{ }}{2^n}^{-1}\)

Example 18: In how many ways the number 210 can n be represented as product of two co–prime numbers?

Solution: The number 210 can be written as  2.3.5.7.

Hence it has 4 distinct prime factors. Required number of ways = \({2^{4-1}} = {\rm{ }}8\) 

Example 19: If \(x\) and \(y\) are co–prime to each other, find the number of solutions of the equation 

\(xy = 2310\).

Solution: The problem is similar to the one discussed above. But in this case the pairs \((a, b)\) and \((b, a)\)   are different. 2310 can be written as 2.3.5.7.11

So total number of pairs = \(2.\left[ {{2^{5-1}}} \right]{\rm{ }} = 32\) ways . 

[ Every pair has to be counted twice]